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What is the change of variables using differentials trick K&K are referring to here?

http://books.google.com.au/books?id...f variables differentials intractable&f=false

(about halfway down the page)

Are there any formalities behind this?

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Also, when people derive the kinematic equations using calculus? I notice they rely on differentials

e.g.

http://physics.info/kinematics-calculus/

The first one, they had a=dv/dt then multiplied both sides by dt and integrated with respect to that variable...perhaps it's cause I'm still not all that comfortable with playing around with differentials like that yet but it doesn't seem 'proper' to do that. Are there alternate methods that DON'T involve treating differentials like that?

Another method that cancelled the differentials is shown here at the end:

I'm not sure about that either

http://books.google.com.au/books?id...f variables differentials intractable&f=false

(about halfway down the page)

Are there any formalities behind this?

---

Also, when people derive the kinematic equations using calculus? I notice they rely on differentials

e.g.

http://physics.info/kinematics-calculus/

The first one, they had a=dv/dt then multiplied both sides by dt and integrated with respect to that variable...perhaps it's cause I'm still not all that comfortable with playing around with differentials like that yet but it doesn't seem 'proper' to do that. Are there alternate methods that DON'T involve treating differentials like that?

Another method that cancelled the differentials is shown here at the end:

I'm not sure about that either

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